Replays work fine for me in Practice and Race. Qualifying doesn't save replays anymore as per patch notes.
From patch notes thread ..
Game:- Fixed issue with position bar showing wrong time left e.g. 30 mins on 60 min race. - Changed so that only race and practice (track test) session replays are saved.
- Replay is saved when the session ends in Single Player.
- Replay is saved when player receives session results from dedicated server in Multiplayer.
- Fixed Replay menu appearing broken.
- Fixed issue with being unable to start championship in R3E if race length was set to 10 minutes.
- Fixed an inconsistency in length between practice, qualifying and race sessions.
- Changed so that the crash dump window shows our name correctly: Sector3 instead of Sector 3.
- Temporarily fixed randomly assigned tyre compounds in MP by making prime default. (Temporary until garage phase is implemented).
- Fixed a small issue with tyre spring calculation being wrong as it was sometimes using the wrong tyre compound.